In what ways can connectionist A.I. be integrated with GOFAI?
A neural net with even a single hidden layer is capable of Universal function approximation - it can approximate any continuous function 'as closely as you like'.

Hence, one option would be to look for GOFAI applications that would benefit from this property - for example, in state-space search approaches where the utility of a state is not readily defined in advance, and could instead be learned.